Ehresmann semigroups were introduced in [1]} as generalizations of inverse semigroups.
We recall their definition here.
An Ehresmann semigroup
is a semigroup \(S\)  with a distinguished subset \(U \subseteq \mathsf {E}(S)\)  of the set of all idempotents,
called the set of projections, equipped with two functions \(\ast , + \colon S \rightarrow U\)  satisfying the following four axioms:
